#### Volume 4, issue 1 (2004)

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The boundary-Wecken classification of surfaces

### Robert F Brown and Michael R Kelly

Algebraic & Geometric Topology 4 (2004) 49–71
 arXiv: math.AT/0402334
##### Abstract

Let $X$ be a compact $2$-manifold with nonempty boundary $\partial X$ and let $f:\phantom{\rule{0.3em}{0ex}}\left(X,\partial X\right)\to \left(X,\partial X\right)$ be a boundary-preserving map. Denote by $M{F}_{\partial }\left[f\right]$ the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to $f$. The relative Nielsen number ${N}_{\partial }\left(f\right)$ is the sum of the number of essential fixed point classes of the restriction $\stackrel{̄}{f}:\phantom{\rule{0.3em}{0ex}}\partial X\to \partial X$ and the number of essential fixed point classes of $f$ that do not contain essential fixed point classes of $\stackrel{̄}{f}$. We prove that if $X$ is the Möbius band with one (open) disc removed, then $M{F}_{\partial }\left[f\right]-{N}_{\partial }\left(f\right)\le 1$ for all maps $f:\phantom{\rule{0.3em}{0ex}}\left(X,\partial X\right)\to \left(X,\partial X\right)$. This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If $X$ is the disc, annulus or Möbius band, then $X$ is boundary-Wecken, that is, $M{F}_{\partial }\left[f\right]={N}_{\partial }\left(f\right)$ for all boundary-preserving maps. If $X$ is the disc with two discs removed or the Möbius band with one disc removed, then $X$ is not boundary-Wecken, but $M{F}_{\partial }\left[f\right]-{N}_{\partial }\left(f\right)\le 1$. All other surfaces are totally non-boundary-Wecken, that is, given an integer $k\ge 1$, there is a map ${f}_{k}:\phantom{\rule{0.3em}{0ex}}\left(X,\partial X\right)\to \left(X,\partial X\right)$ such that $M{F}_{\partial }\left[{f}_{k}\right]-{N}_{\partial }\left({f}_{k}\right)\ge k$.

##### Keywords
boundary-Wecken, relative Nielsen number, punctured Möbius band, boundary-preserving map
##### Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 54H25, 57N05